Thanks @hpi, seems that we posted both at the same time.
But your answer is correct. The discrepancy in the paths in my answer was the result of the parameter build_path in my ESPHome YAMLs. It was set to the same name as the node name, which made sense long time ago. I have removed it now.
Thanks again.
Maybe some of you pros might also be interested in having a look at the following discussion that I started. It is somewhat related and I am clearly approaching the limits of my Linux knowledge over there.